Natural continued fraction

Question

My question concerns the following sequence :

$$u_0=0+\frac{1}{2} \quad u_1 =0+\frac{1}{2+\frac{3}{4}} = \frac{4}{11} \quad u_2=0+\frac{1}{2+\frac{3}{4+\frac{5}{6}}}=\frac{29}{76} \quad u_3=0+\frac{1}{2+\frac{3}{4+\frac{5}{6+\frac{7}{8}}}}=\frac{52}{137}$$

It is distinct from the harmonic continued fraction : Convergence of a Harmonic Continued Fraction

It can be defined by the recurrence relation : $$v_n=2n+1-\frac{1}{2n+2} \qquad v_{k}=2k+\frac{2k+1}{v_{k+1}} \quad (0\leq k <n) (*)$$

Applying the formula (*) $n$ times, we have $u_n=v_0$.

A simple Python script gives the following results : $$ \frac{1}{2}, \frac{4}{11}, \frac{29}{76}, \frac{52}{137}, \frac{2861}{7534}, \frac{37192}{97943}, \frac{557881}{1469144} $$ The sequence seems to converge to $\ell\approx 0.37973195474099564$

from fractions import Fraction
def f(n):
    v = Fraction(2 * n + 1, 1) - Fraction(1, 2 * n + 2)
    for i in range(n - 1, -1, -1):
        v = Fraction(2 * i, 1) + Fraction(2 * i + 1, v)
    return v

print([f(k) for k in range(10)])

Here is my question : what is the exact value of $\ell$ and how can we prove it?

Answer 1

According to sequence A113014 of the OEIS, $$\ell = \frac{\sqrt{2e/\pi}}{\operatorname{erfi}{(1/\sqrt{2})}} - 1, $$

where $\operatorname{erfi}(\cdot)$ is the imaginary error function.

I don't know how to prove this, yet.

Answer 2

This is an addendum to Muller's answer. Since he found out that the OP's cfrac $\ell$ involved the error function $\operatorname{erf}(x)$, we can connect it to Ramanujan's beautiful identity,

$$\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots+\,\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\ddots}}}}$$

The closed-form of its infinite series part is,

$$\sum_{n=0}^\infty \prod_{k=0}^n\frac{x^n}{2k+1} = \sum_{n=0}^\infty \frac{x^n}{(2n+1)!!} = \sqrt{\frac{\pi\,e^x}{2x}} \operatorname{erf}\Big(\sqrt{\tfrac x 2}\Big)$$

Therefore at $x=-1$ we have the imaginary error function,

\begin{align}\sqrt{\frac{\pi}{2\,e}} \operatorname{erfi}\Big(\sqrt{\tfrac 1 2}\Big) &= 1-\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}-\frac{1}{1\cdot3\cdot5\cdot7}+\dots\\ &= \frac{1}{\color{red}1+\ell} = \cfrac1{\color{red}1+\cfrac{1}{2+\cfrac{3}{4+\cfrac{5}{6+\ddots}}}}\\ &=0.724778459007076331\dots \end{align}

which is OEIS A306858.